1
general::
Slight or no consolation. For example, He can't lend us his canoe but will tell us where to rent one? that's cold comfort. The adjective cold was being applied to comfort in this sense by the early 1300s, and Shakespeare used the idiom numerous times.
American Heritage Idioms
2
general::
Phrase(s): cold comfort
no comfort or consolation at all. • She knows there are others worse off than her, but that’s cold comfort. • It was cold comfort to the student that others had failed as he had done.
McGrawhill's American Idioms And Phrasal Verbs